Polarization-varying anisotropic terahertz microspectroscopy (PV-ATM) measures polarization-resolved terahertz absorption fingerprints of single protein and RNA crystals, and it is sensitive to inhibitor binding and other functional-state changes. However, the measured spectra cannot currently be interpreted at the molecular level: no existing modeling framework can predict the frequency-dependent anisotropic dielectric response (both anisotropic absorption and the birefringent refractive index) of a biomolecular crystal from its structure, and it remains undetermined whether terahertz circular dichroism contributes to the measured electro-optic signal alongside birefringent polarization rotation. The open problem, stated explicitly by Niessen et al. in 2019, is to develop a structure-based quantitative description of the anisotropic THz dielectric response of protein and RNA crystals, to settle the possible circular-dichroism contribution, and thereby to assign each observed PV-ATM spectral feature to a specific collective intramolecular vibration.

Background

Proteins and other biomacromolecules undergo collective vibrational motions in which large groups of atoms move in a coordinated way, at frequencies of roughly 0.3–3 THz (about 10–100 cm−1^{-1}). These low-frequency motions are of interest because they are thought to facilitate conformational changes underlying enzyme catalysis, allostery, and ligand binding. Terahertz time-domain spectroscopy (THz-TDS), which uses femtosecond pulses to measure a sample's absorption and refractive index across this frequency band, can in principle observe them, but two obstacles have long blocked interpretable spectra of proteins: liquid water absorbs THz radiation extremely strongly, and a protein's vibrational density of states (VDOS, the distribution of vibrational mode frequencies) is broad and quasi-continuous, so ordinary solution spectra look nearly featureless.

Anisotropic terahertz microscopy (ATM) sidesteps both obstacles by measuring protein microcrystals, where all molecules share one orientation. The collective modes of an oriented molecule absorb THz light only when the electric field of the light points along the mode's dipole-transition direction, so the absorption depends on the polarization of the THz light relative to the crystal axes (linear dichroism, i.e. polarization-dependent absorption). Because the absorption of an anisotropic crystal depends on orientation, one must measure at several relative orientations. In ATM and its successor PV-ATM, the transmitted THz field is detected by electro-optic (EO) sampling: the THz pulse passes through an electro-optic crystal (here ZnTe) together with a near-infrared probe pulse, and the Pockels effect (an applied electric field linearly changes a crystal's birefringence) rotates the probe polarization in proportion to the instantaneous THz field, which is then read out. In early ATM, the sample itself was rotated in the near field — a difficult mechanical requirement making a full 360-degree dataset take about 24 hours. Niessen et al. (2019) introduced polarization-varying anisotropic terahertz microscopy (PV-ATM), in which the sample stays fixed and a rotating polarizer varies the THz polarization instead; this cut acquisition time by roughly a factor of six and, importantly, made the measurement sensitive not only to anisotropic absorption but also to birefringence (the orientation- and frequency-dependent refractive index of the crystal, which rotates the THz polarization as it propagates). They validated the technique on sucrose single crystals, where a full electromagnetic simulation using separately measured anisotropic absorption and refractive-index data (Lorentzian dielectric functions plus Jones-matrix calculus, the standard transfer-matrix treatment of polarization optics) reproduced the measured spectra, and then recorded distinct polarization-resolved fingerprints for hen egg-white lysozyme (CEWL), dihydrofolate reductase (DHFR), photoactive yellow protein (PYP), and an RNA G-quadruplex (a four-stranded nucleic-acid structure formed by stacked guanine tetrads), including strong changes of the lysozyme spectrum upon binding the inhibitor tri-acetyl-glucosamine (3NAG), with the free-lysozyme resonances near 26, 46, and 53 cm−1^{-1} replaced by resonances near 16, 24, 31, 41, and 51 cm−1^{-1}.

Interpreting such fingerprints requires assigning each spectral feature to a specific collective motion. The natural tool is normal mode analysis (NMA), in which the harmonic force-constant matrix built from a molecular mechanics force field is diagonalized to yield the molecule's collective vibration frequencies and displacement patterns. For small molecular crystals (e.g. sucrose or amino acids), first-principles density-functional theory with periodic boundary conditions, combined with anisotropic THz spectroscopy, has achieved reliable mode assignment (Singh et al., 2012). Protein crystals are far too large for that route, and empirical-force-field NMA has reproduced protein anisotropic absorption bands only qualitatively. The sucrose simulations in Niessen et al. showed, moreover, that quantitative PV-ATM simulation requires the anisotropic refractive index, not just the anisotropic absorbance: the extra spectral structure that makes PV-ATM fingerprints so distinctive arises substantially from resonant birefringence and from the polarization dependence of the EO detection. The authors state that current protein modeling cannot predict this level of frequency-dependent anisotropic dielectric response, which makes precise assignment of PV-ATM features out of reach. A second, entangled unknown is circular dichroism (CD, differential absorption of left- and right-circularly polarized light): chiral biomolecular crystals could in principle show CD resonances at THz frequencies — terahertz optical activity has been observed in sucrose single crystals (Kröll et al., 2007) — and the EO detection geometry of PV-ATM might be sensitive to it, so it is not yet known how much of the measured signal, if any, is circular dichroism rather than birefringent polarization rotation. Both questions concern the same measured spectra and together define the missing quantitative, structure-based interpretation of PV-ATM. (The same paper also lists two further directions — realizing molecular orientation control in solution, e.g. by electrostatic or laminar-flow alignment in microfluidic devices, and testing the functional relevance of candidate modes by resonant excitation while monitoring ligand binding or catalytic rates; these are applications enabled by, but distinct from, the quantitative interpretation problem stated here.)

Problem Statement

Develop a quantitative, structure-based interpretation of polarization-varying anisotropic terahertz microspectroscopy (PV-ATM) of biomolecular crystals, as requested by Niessen et al. (2019), consisting of three jointly supporting parts with one common objective — converting PV-ATM dynamical fingerprints into specifically identified collective motions:

  1. A method that predicts, from a protein or RNA crystal structure (space group, molecular orientation, and hydration state), the full frequency-dependent anisotropic dielectric response in the THz range (roughly 10–100 cm−1^{-1}) — both the polarization-dependent absorption coefficient and the birefringent refractive index along the relevant crystal axes — with accuracy sufficient for quantitative simulation of PV-ATM spectra through the electro-optic-detection forward model (Jones-matrix propagation including the EO crystal's polarization-dependent response), as demonstrated for sucrose in the source paper but without using measured THz spectra as fitting input.

  2. A determination — by the validated model, by dedicated polarization-resolved measurement, or both — of whether terahertz circular dichroism contributes to the measured PV-ATM electro-optic signal of biomolecular crystals, in addition to anisotropic absorption and birefringent polarization rotation.

  3. Using 1 and 2, an assignment of the measured PV-ATM spectral features to specific collective intramolecular vibrations (normal modes), for example identifying which collective motions of hen egg-white lysozyme give rise to the resonances near 26, 46, and 53 cm−1^{-1} that shift or reorient upon binding the inhibitor tri-acetyl-glucosamine, with the assignments checked for consistency with the measured frequency, polarization, and angular dependence.

The problem is solved when simulations constructed without fitted THz spectral input reproduce measured protein or RNA crystal PV-ATM spectra to the quantitative standard of the sucrose validation in the source, the circular-dichroism question is settled, and the principal spectral features of at least one benchmark biomolecular crystal (e.g. lysozyme, DHFR, PYP, or the RNA G-quadruplex) are thereby assigned to identified collective motions. The source's further aspirations — orientation control for solution-phase anisotropic measurements and functional-relevance tests by resonant excitation — are adjacent applications of such an assignment, not part of this completion boundary.

References

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